mlsanityData to 7 Oct 2026

Bayesian Quadrature

arXiv 2602.16218 ↗First posted 18 Feb 2026cs.LG cs.NA math.NAcited by 1

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Abstract (arXiv)

Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 63 of its 201 references that are in the map. First posted on arXiv 18 Feb 2026.

Builds on

63 of 201 references.

References · 63

  1. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 360
  2. 2018When Gaussian Process Meets Big Data: A Review of Scalable GPscited by 205
  3. 2015Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)cited by 175
  4. 2015On the Equivalence between Kernel Quadrature Rules and Random Feature Expansionscited by 136
  5. 2014Control functionals for Monte Carlo integrationcited by 59
  6. 2015Probabilistic Numerics and Uncertainty in Computationscited by 59
  7. 2017Bayesian Probabilistic Numerical Methodscited by 50
  8. 2014Sampling for Inference in Probabilistic Models with Fast Bayesian Quadraturecited by 31
  9. 2020Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothnesscited by 29
  10. 2015Probabilistic Integration: A Role in Statistical Computation?cited by 26
  11. 2019Kernel quadrature with DPPscited by 26
  12. 2021Positively Weighted Kernel Quadrature via Subsamplingcited by 25
38 more
  1. 2018Bayesian Optimization with Expensive Integrandscited by 24
  2. 2021Emulation of physical processes with Emukitcited by 24
  3. 2014Probabilistic ODE Solvers with Runge-Kutta Meanscited by 22
  4. 2019Convergence Guarantees for Adaptive Bayesian Quadrature Methodscited by 22
  5. 2020Kernel interpolation with continuous volume samplingcited by 21
  6. 2015Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guaranteescited by 20
  7. 2022Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombinationcited by 20
  8. 2019A Modern Retrospective on Probabilistic Numericscited by 18
  9. 2019Active Multi-Information Source Bayesian Quadraturecited by 17
  10. 2023Sampling-based Nystr"om Approximation and Kernel Quadraturecited by 17
  11. 2017Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settingscited by 16
  12. 2022Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posedcited by 16
  13. 2016Convergence guarantees for kernel-based quadrature rules in misspecified settingscited by 15
  14. 2018Bayesian Quadrature for Multiple Related Integralscited by 15
  15. 2020Distributionally Robust Bayesian Quadrature Optimizationcited by 15
  16. 2020Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functionscited by 15
  17. 2018Improving Quadrature for Constrained Integrandscited by 12
  18. 2023Kernel quadrature with randomly pivoted Choleskycited by 11
  19. 2018A Bayes-Sard Cubature Methodcited by 10
  20. 2023Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Samplingcited by 10
  21. 2016Active Uncertainty Calibration in Bayesian ODE Solverscited by 9
  22. 2020Semi-Exact Control Functionals From Sard's Methodcited by 9
  23. 2025A Dictionary of Closed-Form Kernel Mean Embeddingscited by 9
  24. 2018Symmetry Exploits for Bayesian Cubature Methodscited by 8
  25. 2022Multilevel Bayesian Quadraturecited by 8
  26. 2022Bayesian Model Selection of Lithium-Ion Battery Models via Bayesian Quadraturecited by 8
  27. 2024Conditional Bayesian Quadraturecited by 8
  28. 2017On the Sampling Problem for Kernel Quadraturecited by 7
  29. 2020Adaptive quadrature schemes for Bayesian inference via active learningcited by 6
  30. 2021Performance analysis of greedy algorithms for minimising a Maximum Mean Discrepancycited by 6
  31. 2025Conformal Prediction as Bayesian Quadraturecited by 6
  32. 2016Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Modelscited by 5
  33. 2017Optimal Monte Carlo integration on closed manifoldscited by 5
  34. 2019Automated Model Selection with Bayesian Quadraturecited by 5
  35. 2020Bayesian Probabilistic Numerical Integration with Tree-Based Modelscited by 5
  36. 2020Bayesian Quadrature Optimization for Probability Threshold Robustness Measurecited by 5
  37. 2021Bayesian Quadrature on Riemannian Data Manifoldscited by 5
  38. 2022A Note on Monte Carlo Integration in High Dimensionscited by 5

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Citing papers per year: 2026: 1.

1 paper.

  1. 2026Hierarchical Bayesian Quadraturecited by 0